# Binary Call Option Theta

## The Binary Call Option Theta measures the change in the price of a binary call option over time and is the gradient of the slope of the binary options price profile versus time decay.

This section on binary call option theta, as with the binary put option theta section, is in two parts:

i.    the first section covers the derivation of the formula (which can be found immediately above the Summary) from first principles, plus the binary call options theta with respect to time to expiry and implied volatility,

ii.    while the second section analyses the theta as reflected by the formula as a useful analytical tool, discusses its drawbacks and provides an alternative ‘practical’ theta, followed by the formula.

### Binary Call Option Theta and Finite Theta

The theta ϴ of any option is defined by:

ϴ = δP / δt

where:

P     =     price of the option

t      =     time in years to expiry

δP     =    a change in the value of P

δt     =    a change in the value of t

N.B. The equation for the binary call options theta can be found at the bottom of the page.

Figure 1 shows binary call option price profiles at different times to expiry. Figure 2 shows how with seven static underlying prices, the binary call options change in value as the days to expiry fall from 25 to 0, so in effect a profile from Figure 2 is a vertical cross section at that underlying price in Figure 1.

When the underlying price is 100.00 the option is at-the-money and the passing of time has no effect on the price of the binary option as it is always 50. When the underlying price is above 100.00 the price profiles all slope upwards reflecting a positive theta, whereas the out-of-the-money profiles, i.e. where S < 100.00, the price profiles all slope down meaning a negative theta.

Fig.1 – Binary Call Option Price profiles w.r.t. Time to Expiry

Fig.2 – Binary Call Option Price profiles w.r.t. Time to Expiry

The theta (as represented by the above formula) measures the gradient of the slopes in Figure 2. When there is over 20 days to expiry price decay (whether negative or positive) is very low; as time passes the theta increases in absolute value with that increase dependent on how close to the strike the underlying is.

Figure 3 is the S=99.75 price profile over the last 11 days of its life. Chords have been added centred around five days to expiry so that, for example, the five-day chord stretches from 7.5 days to 2.5 days to expiry. Since the price profile is decreasing exponentially, the gradient of the chords decrease the longer the length of the chord.

The gradient of the chord is defined by:

Gradient     =     ‒ ( P2 – P1 ) / ( t2 – t1 )

where:

t2    =    t + δt

t1    =    t – δt

P2    =     Binary Call value at t2

P1    =    Binary Call value at t1

i.e.            Gradient    =          ― (37.3446 ― 16.9094) / (9 ‒ 1)      =    ― 2.5544

Fig.3 – Slope of the Theta at \$99.75 plus approximating Theta ‘chords’

as indicated in the bottom row of the central column of Table 1.

The gradients of the ‘5 day chord’ and ‘2 day  chord’ are calculated in the same manner and are also presented in the central column of Table 1.

Table 1 - From Gradient of Chord to Call Theta
Days to Expiry9.07.56.05.04.02.51.0
δt=0-1.5446
δt=234.6912-1.579831.5315
δt=536.2109-1.803527.1933
δt=837.3446-2.554416.9094

Table 1

As the time difference narrows (as reflected by δt = 5 and δt = 2) the gradient tends to the theta of ―1.5446 at 5 days to expiry, i.e. where δt = 0. The theta is therefore the first differential of the binary call fair value with respect to time to expiry and can be stated mathematically as:

as δt → 0,     ϴ = dP / dt

which means that as δt falls to zero the gradient approaches the tangent (theta) of the price profile of Figure 2 at 5 days.

### Binary Call Option Theta w.r.t. Time to Expiry

Figure 1 illustrates 5.0% implied volatility binary call profiles with Figure 4 providing the associated thetas for the same days to expiry.

Irrespective of the days to expiry the theta when at-the-money is always zero. When out-of-the-money the binary call theta is always negative (as with out-of-the-money conventional call options) but when in-the-money the binary call options theta is positive (unlike in-the-money conventional call options).

With sufficient days to expiry (25 days in Figure 4) the binary call option theta is almost flat at close to zero. As time passes the absolute maximum value of the theta increases with the peak and trough progressively closing on the strike. This can be explained by the case where there is just 0.5 days to expiry where at an underlying price of 99.90 the binary call option is worth 29.4059 which is the amount that the option will decrease by over the next half-day if the underlying remains at 99.90.

Fig.4 – Binary Call Option ‘Theoretical’ Theta w.r.t. Time to Expiry

Although at 99.90 and 1-day to expiry the binary call option is worth 35.0638 (5.6579 more than at the half-day to expiry) the binary call theta is lower as the theta is an annual measurement, not necessarily a practical one.

### Binary Call Option Theta w.r.t. Implied Volatility

Figures 5 & 6 provide the binary call options price profiles over a range of implied volatilities with the associated binary call theta. As is usual the implied volatility has a similar effect on the price profiles but there are some subtle differences between the binary call theta profiles of Figs. 4 & 6.
The maximum absolute theta in Figure 6 is fairly steady at around 2.43 irrespective of the implied volatility, although the implied volatility does determine how close to the strike the peak and trough in theta is.

Fig.5 – Binary Call Option Price profiles w.r.t. Implied Volatility

Fig.6 – Binary Call Option ‘Theoretical’ Theta w.r.t. Implied Volatility

Irrespective of implied volatility the binary call theta travels through zero for the now familiar reason that at-the-money binaries are priced at 50, or very close to it.

### ‘Theoretical’ Theta and ‘Practical’ Theta

From Figure 3 above it is (hopefully) visually apparent that an equal measure of time backwards provides an increase in call option value which is less than the decrease in option value for an equivalent jump forwards in time, e.g. at time 5 days to expiry the binary call option fair value is 33.3357, so using the example with δt=2, the 6-day and 4-day options are worth respectively 34.6912 and 31.5315. So from the 6th day to the 5th day the option loses:

Price decay from Day 6 to Day 5  =  (34.6912―33.3357) = 1.3555

while from the 5th day to the 4th day the option loses:

Price decay from Day 5 to Day 4  =  (33.3357―31.5315) = 1.8042

Table 2 presents the option value at days to expiry from 7 to 0 with the daily difference plus the ‘theoretical’ theta; it is apparent that the actual decay from one day to the next is greater than the theoretical theta. The ‘theoretical’ binary call theta in this instance is derived from the formula of Eq(1) above divided by 365 (Eq(1) provides an annual rate) and multiplied by 100 (Eq(1) assumes a binary option price range between 0 and 1, not 0 and 100).

Table 2 - Binary Call Option Fair Value with associated day's decay and theta
Days to Expiry76543210
Fair Value35.756934.691233.335731.531528.962924.883516.90940
Day's Decay-1.0658-1.3555-1.8042-2.5687-4.0793-7.9742-16.9094N/A
Theoretical Theta-0.9545-1.1915-1.5446-2.1128-3.1355-5.3449-12.0436N/A

Table 2

This begs the question as to the efficacy of using the formula of Eq(1) when might it not be simpler to compute the theta as calculated from the ‘Day’s Decay’ row of Table 2. Not particularly mathematically elegant, but there are a number of equally inelegant adjustments made by market practitioners to ‘elegant’ mathematical models in order to make them work, with volatility ‘skew’ being one of the more obvious. To be even deeper, the CAPM financial model is dependent on a ‘risk-free’ rate of interest…………is there such a thing as a ‘risk-free’ rate of interest?: what if the IMF was downgraded by Moody’s over the PIGS?!

Figures 7a-f offer graphical illustrations of the difference between ‘theoretical’ theta and ‘practical’ theta, a term I’ve coined to simply describe the actual change in price from one day to the next. Figure 7a shows that as the binary call option price decay (either positive or negative) is negligible then the theoretical theta almost overlaps the practical theta, especially when implied volatility is low.

Fig.7a – Binary Call Option Theta, ‘Theoretical’ & ‘Practical’, 25-Days to Expiry w.r.t. Implied Volatility

With 10 and 4 days to expiry the theoretical theta gradually becomes more inaccurate as a measure of actual option price change with the actual time decay being absolutely greater at the peaks and troughs of the theta binary call options theta profiles but becoming lesser as the underlying moves away from the strike. This ‘smoothing’ is what might be expected when comparing the actual price changes of the ‘practical’ theta and the notional price changes portrayed by the ‘theoretical’ theta which itself is an annualised rate and in effect has a built in averaging mechanism.
The left hand scales of Figures 7a-c are gradually increasing in value as the theta increases over time.

Fig.7b – Binary Call Option Theta, ‘Theoretical’ & ‘Practical’, 10-Days to Expiry w.r.t. Implied Volatility

Fig.7c – Binary Call Option Theta, ‘Theoretical’ & ‘Practical’, 4-Days to Expiry w.r.t. Implied Volatility

When there is one day to expiry (Figure 7d) the undervaluation of time decay as generated by the ‘theoretical’ theta is at its most pronounced because at this point the ‘practical’ theta is in fact the binary call option premium when out-of-the-money and 100 less the binary call option premium when in-the-money.

Fig.7d – Binary Call Option Theta, ‘Theoretical’ & ‘Practical’, 1-Day to Expiry w.r.t. Implied Volatility

Finally Figures 7e & 7f illustrate the absolute ‘theoretical’ theta rising aggressively while the absolute ‘practical’ theta is now falling, the latter due to the lower premium of the option.

Fig.7e – Binary Call Option Theta, ‘Theoretical’ & ‘Practical’, 0.4-Days to Expiry w.r.t. Implied Volatility

Fig.7f – Binary Call Option Theta, ‘Theoretical’ & ‘Practical’, 0.1-Days to Expiry w.r.t. Implied Volatility

The scales of Figures 7e & 7f are worth noting, in particular Fig 7f where the ‘theoretical’ theta now rises above 100, which is an interesting concept since the maximum range of the binary call option is limited to 100!

Points of note are:

1)    Whereas conventional call option thetas are always negative as time value is always positive, time value with binary call options can be positive or negative dependent on whether they are in- or out-of-the-money.

2)    Whereas with conventional call options theta is always at its absolute highest when at-the-money, the binary call options theta when at-the-money is always zero.

3)    Out-of-the-money binary call options have negative or zero theta, in-the-money binary call options have a zero or positive theta.

4)    Using Eq(1) to calculate theta can generate theta in excess of 100.

### Formula

$\textup{Binary&space;Call&space;Option&space;Theta}=re^{-rt}N\left&space;(&space;d_{2}&space;\right&space;)+e^{-rt}{N}'\left&space;(&space;d_{2}&space;\right&space;)\left&space;(&space;\frac{d_{1}}{2t}-\frac{r-D}{\sigma&space;\sqrt{t}}&space;\right&space;)$

where:

$d_{1}=\frac{log\left&space;(&space;\frac{S}{E}&space;\right&space;)+\left&space;(&space;r-D+\frac{\sigma&space;^{2}}{2}&space;\right&space;)t}{\sigma&space;\sqrt{t}}$

$d_{2}=\frac{log\left&space;(&space;\frac{S}{E}&space;\right&space;)+\left&space;(&space;r-D-\frac{\sigma&space;^{2}}{2}&space;\right&space;)t}{\sigma&space;\sqrt{t}}$

${N}'\left&space;(&space;x&space;\right&space;)=\frac{1}{\sqrt{2\pi&space;}}.e^{-0.5x^{2}}$

and:

$\textup{S}=\textup{price&space;of&space;the&space;underlying}$

$\textup{E}=\textup{strike/exercise&space;price}$

$\textup{r}=\textup{risk&space;free&space;rate&space;of&space;interest}$

$\textup{D}=\textup{continuous&space;dividend&space;yield&space;of&space;underlying}$

$\textup{t}=\textup{time&space;in&space;years&space;to&space;expiry}$

$\sigma&space;=\textup{annualised&space;standard&space;deviation&space;of&space;asset&space;returns}$

N.B.
(i)    The theta generated by the above equation is an annualised number, so should a daily theta be required as an approximation then the theta needs to be divided by 365.

(ii)    This formula is based on binary call option prices that range between 0 and 1. Should a theta be required for binary call option prices that range between 0 and 100 then the theta should be multiplied by 100.

### Summary

If theta is solely represented by the results of Eq(1) then it is a useful tool for establishing daily time decay if divided by 365 plus there is sufficient time to expiry. But as time to expiry falls this ‘theoretical’ theta becomes increasingly inaccurate as a tool for forecasting the binary option price change over time.

The delta can be hedged away by trading the underlying; until time itself becomes a tradable entity (a future?) hedging theta can only be achieved by trading other options.

As with deltas, as expiry approaches the theta can reach ludicrously high numbers so one should always observe the tenet: “Beware Greeks bearing silly analysis numbers…” (as ever).

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